Abstract
This paper discusses Michael Dummett’s attempt to base the use of intuitionistic logic in mathematics on a proof-conditional semantics. This project is shown to face significant obstacles resulting from the existence of variants of standard intuitionistic logic. In order to overcome these obstacles, Dummett and his followers must give an intuitionistically acceptable completeness proof for intuitionistic logic relative to the BHK interpretation of the logical constants, but there are reasons to doubt that such a proof is possible. The paper concludes by proposing an alternative way of thinking about why one should use intuitionistic logic when doing mathematics.
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Notes
The term ‘constructive mathematics’ is ambiguous. In this paper, it will be used to refer to any approach to mathematics that makes use of intuitionistic logic. So understood, constructive mathematics might be motivated by a view that mathematical objects are constructed in some sense, but it does not have to be. In the last section of this paper, we will examine the relationship between traditional constructivist ontologies and the use of intuitionistic logic in mathematics.
The acronym contains the initials of Brouwer, Heyting, and the Russian mathematician Andrei Kolmogorov. Brouwer himself never explicitly articulated anything like the BHK interpretation, but Heyting’s work on logic was initially an attempt to formalize the reasoning characteristic of his teacher’s intuitionistic mathematics. Kolmogorov had presented something similar to Heyting’s semantics a little earlier in Kolmogorov (1932).
See Dummett (1975) for a well known presentation of it.
Constructivists reject the full-strength Axiom of Choice, but not on grounds of intuitive clarity. Rather, the problem with the axiom for constructivists is that it entails the validity of the LEM.
Heyting, who tends to be more ecumenical and pragmatic, may be open to holistic justifications, but he does not appeal to them in his arguments for various logical laws.
We have modified the notation in this passage to conform to our standards rather than the ones that Heyting uses in his book.
Elsewhere (Dummett 1991, p. 295), Dummett suggests taking absurdity to be equivalent to the conjunction of every atomic sentence of the language. The problem with this is that, in a suitably restricted mathematical language, this conjunction will not always be absurd. For instance, if our language contains only symbols for equality and the number 1, then the only atomic formula is ‘\(1=1\)’, which is not absurd. The reason why the maximal atomic conjunction is absurd in the language of ordinary arithmetic is because we know in advance that, say, 1 is not 2, so including ‘\(1=2\)’ as a conjunct makes the whole conjunction false. The point is that this will not allows us to avoid smuggling negation back into our understanding of absurdity. See Cook and Cogburn (2000) for more on this.
Dummett describes some of these formal systems in Dummett (2000, ch. 4).
For this reason, it is a bit of a distortion to present Heyting as a precursor to Dummett, as we have done in this paper. Our reason for introducing this distortion earlier was to emphasize that the BHK interpretation has been taken to be an important part of constructive mathematics even before Dummett, even though he more than anybody else has been responsible for tying the mathematics to a particular theory of meaning.
Other parts of Brouwer’s work, such as the theory of choice sequences, call for non-logical explications. Troelstra, in particular, has done much to carry out this sort of explication in Troelstra (1977) and elsewhere.
This is reflected in the sequence of topics in Dummett’s Elements of Intuitionism, which begins with a discussion of the meanings of the logical constants and proceeds to a chapter on intuitionistic logic before turning to the mathematics itself.
This claim is based on the author’s personal experience and conversations with other non-Dummettian proponents of constructive mathematics.
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Acknowledgments
Earlier versions of this paper were presented at California State University, San Bernardino and the University of Western Ontario. The author is grateful to the philosophy departments at those institutions for useful discussions. Thanks are also due to two anonymous reviewers, Matt Carlson, David Fisher, Gary Ebbs, Neil Tennant, and especially Charles McCarty for fruitful feedback.
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Koss, M.R. Some Obstacles Facing a Semantic Foundation for Constructive Mathematics. Erkenn 80, 1055–1068 (2015). https://doi.org/10.1007/s10670-014-9697-7
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DOI: https://doi.org/10.1007/s10670-014-9697-7